One of the world’s most popular books for children and adults

1. "the shortcomings of all mankind" is not a masterpiece by Jonathan Swift. The modifier "a masterpiece by Jonathan Swift" should either modify Gulliver's Travels or be linked to it by a linking verb. Hence A is incorrect.

2. The first thing I observed here was the comma that was just before the underline started. Hence, if the modifier after the comma is intended to modify Gulliver's Travels, then the verb publish should be strictly a past-participle (Recall that present participle modifies the preceding noun only when there is no comma). This eliminates B and D.

3. Between C and E, the difference is that of a meaning issue. While the book may have been written to mock the shortcomings of all mankind, I cannot claim that from the statement. Hence, E is incorrect.

4. Now if I replace the underlined part with C, I get the following:
One of the world’s most popular books for children and adults alike, Gulliver’s Travels, published anonymously in 1725, mocks the shortcomings of all mankind and is a masterpiece by Jonathan Swift, an Irish author who in his early writings ridiculed shallow thinkers and supported the merits of ancient classics.
Here the modifiers are all placed correctly and the core sentence is a complete thought: Gulliver’s Travels mocks the shortcomings of all mankind and is a masterpiece by Jonathan Swift. "mocks" and "is" are both verbs and hence the parallel structure is also taken care of. So I think C is the perfect option.